24.111 Philosophy of QM Spring, 2005
Assignment #3: Bell’s Inequalities and spin measurements (due 2/28/05)
One point each, except for problem 2, which is worth 2 points. It would be a good idea to read
Albert, chapters 2 and 3, and to review section VI of handout #5 (tensor products).
1. Describe, in the abstract, what would be required of an experiment performed on a spin-1/2
particle in the (UP, 0°) state, in order for that experiment to count as a simultaneous quantum
mechanical measurement of spin in two (arbitrary) distinct directions, θ1 and θ2. Note that
the outcome probabilities need to conform to the statistical algorithm, but that it is not
required that this simultaneous measurement “collapse” the state of the particle. Note also
that the words “in the abstract” are there for a reason: in particular, you will be going down a
blind alley if you try to imagine some configuration of Stern-Gerlach magnets that would do
the job. Finally, you should ignore the fact that, according to standard quantum mechanical
lore, such an experiment is impossible. (Part of the point of this and the next exercise is to
show one important reason for this part of quantum mechanical lore.) Hint: How many
distinguishable outcomes would this experiment need to have? What would their
probabilities have to be?
2. Show that, given the quantum mechanical probabilities for the experimental arrangement
described in handout #2, it is not possible to design an apparatus that simultaneously
measures the spin of particle 1 in both the 0° direction and the +120° direction (i.e., the
probabilities that such a joint measurement yields for UP,0° and UP,+120° would have to be
the same as those yielded, respectively, by an ordinary 0°-measurement and by an ordinary
+120°-measurement). You may assume that if you could design such an apparatus, then you
could also design an apparatus which simultaneously measures the spin of particle 2 in both
the 0° direction and the -120° direction. Hint: It’s easy—just think about the derivation of
Bell’s Inequalities in handout #2.
3. It is possible to prepare a spin-1/2 particle in a state such that no matter what spinmeasurement is performed on it, Prob(UP) = .5. Give a convincing argument that this state
cannot be represented by a vector.
4. Consider a system consisting of two spin-1/2 particles, one in the x-up spin state and the
other in the x-down spin state (x-up = certain to go up through a 90°-magnet; x-down =
certain to go down through a 90°-magnet). Then the state of the system as a whole (well, the
spin part of that state) can be represented by the vector x+⊗ x-. (The “⊗” symbol is for the
tensor product—what Albert calls a kind of “vector multiplication”.) Write this vector as a
linear combination of the vectors z+⊗z+, z+⊗z-, z-⊗z+, and z-⊗z-. That is, find the
coefficients a, b, c, and d in the equation
x+⊗x- = a z+⊗z+ +b z+⊗z- + c z-⊗z+ + d z-⊗z-.
It will help to remember that x+ = 1/√2 (z+ + z-), and x- = 1/√2 (z+ - z-).
5. Now do the same for the vector x-⊗x+.
6. Use the results of the last two problems to write the vector 1/√2(x+⊗x- - x-⊗x+) as a linear
combination of the vectors z+⊗z+, z+⊗z-, z-⊗z+, and z-⊗z-.